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FINITELY RAMIFIED GRAPH-DIRECTED FRACTALS, SPECTRAL ASYMPTOTICS AND THE MULTIDIMENSIONAL RENEWAL THEOREM

Published online by Cambridge University Press:  27 January 2003

B. M. Hambly  and
S. O. G. Nyberg
B. M. Hambly
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford OX1 3LB, UK ([email protected])
S. O. G. Nyberg
Affiliation:
Computas AS, Vollsveien 9, PO Box 482, 1327 Lysaker, Norway ([email protected])
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Abstract

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We consider the class of graph-directed constructions which are connected and have the property of finite ramification.By assuming the existence of a fixed point for a certain renormalization map, it is possible to construct a Laplaceoperator on fractals in this class via their Dirichlet forms. Our main aim is to consider the eigenvalues of theLaplace operator and provide a formula for the spectral dimension, the exponent determining the power-law scaling inthe eigenvalue counting function, and establish generic constancy for the counting-function asymptotics. In order to dothis we prove an extension of the multidimensional renewal theorem. As a result we show that it is possible for theeigenvalue counting function for fractals to require a logarithmic correction to the usual power-law growth.

AMS 2000 Mathematics subject classification: Primary 35P20; 58J50. Secondary 28A80; 60K05; 31C25

Keywords

fractalDirichlet formcounting-function asymptoticsrenewal theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 46 , Issue 1 , February 2003 , pp. 1 - 34
Copyright
Copyright © Edinburgh Mathematical Society 2003